Using tensor networks as machine learning models
I do not feel comfortable proclaiming that I would be a machine learning expert.
But I want to point out that there is indeed interest in applying tensor networks in machine learning settings.
Let me highlight three particular settings in the following.
They can be used to find the governing equations behind dynamical systems analogous to the SINDy algorithm.
The reasoning is that the dynamics of your system may not be sparse but may indeed have a low-rank structure. (See this paper for a comparison of sparse and low-rank structures and this paper for a discussion about why low-rank matrices so often appear in big data.)
The resulting modification of SINDy is the appropriately named MANDy algorithm.
But if your dynamical law exhibits additional structure (despite being low-rank) then this structure is not used. (Requiring you to collect unnecessary samples.)
This work tackles this issue by allowing you to specify the structure that you expect your law to exhibit.
Another area of interest is the application of tensor networks as a means of compression.
Tensor networks can be used to approximate multivariate functions.
Either directly (see this article or this article for further details) or by representing their high-dimensional coefficient tensor (see e.g. here).
The last paper also highlights where such functions may occur: when solving stochastic or parametric PDEs.
&quot;What does this have to do with machine learning?&quot; you may rightfully ask.
These PDEs are hard to solve and it is often easier to minimize a residual than to apply a Galerkin method. The results can be shown to be equivalent with high probability. (See this paper for a proof of this statement and an application to some stochastic PDEs or this paper for an application to an optimal control problem.)
Moreover, there is the problem of tensor completion - a generalization of matrix completion to higher dimensions. (This and this are references for algorithms.)
These methods find application in data science where they can be used to decompose or denoise data. (Unfortunately, I don't have a reference with examples but I remember a presentation where these methods where used on EEG signals.)
Additionally, I recently found this interesting symposium covering even more potential applications of tensor networks in machine learning:
https://itsatcuny.org/calendar/quantum-inspired-machine-learning